Approximating k-cuts using Network Strength as a Lagrangean Relaxation

Given an undirected, edge-weighted connected graph, the k-cut problem is to partition the vertex set into k non-empty connected components so as to minimize the total weight of edges whose end points are in different components. We present a combinatorial polynomial-time 2-approximation algorithm for the k-cut problem. We use a La-grangean relaxation (also suggested by Barahona [2]) to reduce the problem to the attack problem, for which a polynomial time algorithm was provided by Cunningham [4]. We prove several structural results of the relaxation, and use these results to develop an approximation algorithm. We provide analytical comparisons of our algorithm and lower bound with two others: Saran and Vazirani [10] and Naor and Rabani [8]. We also provide computational results comparing the performance of our algorithm on random graphs with respect to the lower bound provided by the attack problem as well as an alternate 2-approximation algorithm provided by Saran and Vazirani [10].